Equianharmonic

In mathematics, and in particular the study of Weierstrass elliptic functions, the equianharmonic case occurs when the Weierstrass invariants satisfy g₂ = 0 and g₃ = 1. This page follows the terminology of Abramowitz and Stegun; see also the lemniscatic case. (These are special examples of complex multiplication.)

In the equianharmonic case, the minimal half period ω₂ is real and equal to

[math]\displaystyle{ \frac{\Gamma^3(1/3)}{4\pi} }[/math]

where [math]\displaystyle{ \Gamma }[/math] is the Gamma function. The half period is

[math]\displaystyle{ \omega_1=\tfrac{1}{2}(-1+\sqrt3i)\omega_2. }[/math]

Here the period lattice is a real multiple of the Eisenstein integers.

The constants e₁, e₂ and e₃ are given by

[math]\displaystyle{ e_1=4^{-1/3}e^{(2/3)\pi i},\qquad e_2=4^{-1/3},\qquad e_3=4^{-1/3}e^{-(2/3)\pi i}. }[/math]

The case g₂ = 0, g₃ = a may be handled by a scaling transformation.